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Week 6 Book Notes - Thermal Physics

by: Hazel Medina

Week 6 Book Notes - Thermal Physics Physics 60

Marketplace > University of California - Irvine > Physics 2 > Physics 60 > Week 6 Book Notes Thermal Physics
Hazel Medina
GPA 3.0
Thermal Physics
Feng, J.

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Hey, everyone! I hope all is well! If you feel a little overwhelmed over thermal physics, then let me help! Download my notes, and keep working hard. :)
Thermal Physics
Feng, J.
Class Notes
thermal physics entropy free energy
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This 3 page Class Notes was uploaded by Hazel Medina on Sunday November 8, 2015. The Class Notes belongs to Physics 60 at University of California - Irvine taught by Feng, J. in Fall 2015. Since its upload, it has received 28 views. For similar materials see Thermal Physics in Physics 2 at University of California - Irvine.


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Date Created: 11/08/15
Physics 60 11022015 Week 6 Ch 63 The Equipartition Theorem 0 Equipartition theorem does not apply to all systems Only applies to systems with energy in the form of quadratic degrees of freedom of the form Eq col2 I c is the constant coefficient and q is any coordinate or momentum variable 0 Consider a system where each q corresponds to independent states where q is discretely spaced by Aq Partition function for this system 2 Zq em Zq e39chA2 Z 1Aq2q e39chAqu and this becomes 2 1Aqfoe39chA2dq Assign x to be x JEq so that dq dxAE which means 2 MAMAE fie e39 is the Gaussian function whose integral cannot be written in terms of elementary functions I Instead use trick that definite integrals from negative infinity to infinity equals xE so that z 1Aq g cat 0 C is the abbreviation for Aq g Average energy of the system is F 1ZoZoB 1cB3912aagcg3912 12B391 12kT Generally equipartition theorem only applies when the spacing between q is much less than kT Ch 64 The Maxwell Speed Distribution 0 Rootmeansquare speed of molecules is vrms 3kTm Can look at relative probabilities of various speeds out of the infinite possible velocities I Pv1 v2 fvv12 Dvdv where Dvis the height of the graph o If the interval between v1 and v2 is infinitesimal then Pvvdv Dvdv Dv is the distribution function Dv oc probability of a molecule having velocity 13 x number of vectors 1 corresponding to speed v I number of vectors 1 corresponding to speed v oc 47w2 0 Using Boltzmann factor we get Dv C4Ttv2e39va22kT proportionality found by using 1 4TtC2kTm32fooO xze39xAzdx We get Maxwell distribution Dv mZTtkT324Ttv2e39va22kT I This gives a parabolic shape where the distribution goes to 0 at v0 where C is the constant of I We can get average speed 17 18kT7l39m which is between vmaxand vrms Ch 65 Partition Functions and Free Energy 0 The number of available microstates for an isolated system with fixed energy U multiplicity QU is the most fundamental statistical quantity of the system The partition function ZT is the most fundamental statistical quantity of a system in equilibrium with a reservoir at temperature T I The Helmholtz free energy is F lenZ or Z e 0 Remember F E U TS and aFamN S which means aFamN F UT Now F lenZ and it becomes aFaT ooTlenZ kan kTooTan Also knowing that B 1kT it becomes ooTan dBdTdoBan UkTZ Then aFaT becomes aFaT knZ kTUkT2 FT UT 0 When T 0 F O kTnZO U0 FO which means F F for all T 0 Knowing F lenZ we find S aFaTVN P orawn and p oFoNTV FkT Ch 66 Partition Functions for Composite Systems 0 How does the partition function for a system of several particles relate to that for each individual particle For example if there are two particles that do not interact with each other and this system has energy E1 E2 then Ztota ZS e39BlEl m5 ZS e39BE1Se39BE2S I If the two particles are distinguishable then Ztota 251 252 o This becomes Ztota 2122 for a system of two noninteracting distinguishable particles 0 Remember that this is only valid for distinguishable particles Ztota 212223ZN for noninteracting distinguishable systems and Ztota 1N21N for noninteracting indistinguishable systems eBE1sleBE252 Ch 67 Ideal Gas Revisited 0 Using 2 1NZlN where 21 is the partition function for one individual molecule we find 21 ZtrZint Knowing Eint is the internal energy rotational vibrational etc and Etr is the molecules tranSIational kinEtiC energy Ztr Ztransitional states eEtrkTand Zint Zinternal states eEintkT 0 Now we will look at how to use the method of counting all independent definiteenergy wavefunctions Remember wavelength is An 2Ln n 1 2 I Using the de Broglie relation p hA to get the magnitude of the momentum we get Pn h An hn2L o This means that the allowed energies for a molecule in a onedimensional box is En pnzZm h2n28mL2 With this we find the partition function to be 21d Zn e39EkT Zn 6 00 A A Unless L andor T IS extremely small then 21d f0 3 h 2 amp28ml 2 zllv 8mLZkTh2 Z kahZL E LLo quantum length to NW hquot2nquot28mLquot2kT o For a molecule moving in 3 dimensions Etr pX22m pf2m pf2m The partition function becomes Ztr ZS e39EkT LxLQLyLQLZLQ Vvo after you sum it up I vo is the quantum volume vo M2 hV Zitka3 I For a singleparticle partition function we get 21 VvoZint Zint is the sum of all relevant internal states 0 The partition function for the entire gas of N molecules is Z 1NVZintVQN The logarithm of the partition function is an NnV anint lnN lnvo 1 0 Predictions Total average energy U 1ZoZoB ooBan which becomes U Uint 32NkT The heat capacity CV oUoT oUintoT 32Nk The Helmholtz free energy is F lenZ NkTan anint lnN lnvo 1 NkTan lnN lnvo 1 Fint o P altawnN NkTV 5 6F6TVN NkllanNVQ 52 dFintdT 39 l1 dFthN 39kTnVZint NVQ


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